The power generated by an electrical circuit (in watts) as a function of its current $x$ (in amperes) is modeled by $P(x)=-15x(x-8)$ What current will produce the maximum power?
The circuit's power is modeled by a quadratic function, whose graph is a parabola. The maximum power is reached at the vertex. So in order to find the current that will produce the maximum power, we need to find the vertex's $x$ -coordinate. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} P(x)&=0 \\\\ -15x(x-8)&=0 \\\\ \swarrow &\searrow \\\\ -15x=0\text{ or }&x-8=0 \\\\ x={0}\text{ or }&x={8} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({8})}{2}=\dfrac82=4$ In conclusion, the maximum power occurs when the current is $4$ amperes.